Effect of Hunting on Red Deer

Modelling Fecal Cortisol Metabolites

Nikolai German, Thomas Witzani, Ziqi Xu, Zhengchen Yuan, Baisu Zhou

Dr. Nicolas Ferry - Bavarian National Forest Park / Daniel Schlichting - StabLab

31 Jan 2025

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

The Goal

assess red deer stress response towards hunting activities

– on 41 individual collared red deer

– within Bavarian Forest National Park

– using FCMs

The Terminology

  • FCMs: Faecal Cortisol Metabolites - a non-invasive method to measure stress through faecal samples

  • collared deer: red deer wearing a GPS-collar, which provides hourly location information

  • Euclidean Distance: Also known as \(L^2\) Distance. Reduces to Pythagorean Theorem for two Dimensions: \[d_{x,y} = ((x_1 - y_1)^2 + (x_2 - y_2)^2)^\frac{1}{2} \\ x,y \in \mathbb{R}^2 \]

The FCMs

  • FCM values do not represent stress level when defecating
  • we expect higher FCM levels when a Deer was stressed sometime within the near past1
  • Reason: gut retention time

Huber et al (2003)

The Approach

  • model FCM levels on spatial and temporal distance to hunting activities

  • Expectation: FCM levels higher when closer in time and space

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

The Datasets

  • Movement: contains the location and datetime1 of the 41 collared deer in the period Feb 2020 - Feb 2023. In total approx. 740 000 observations2

  • Hunting Events: contains location and date of hunting events in the National Park - in total 1270 events, 890 of them with full timestamp

  • FCM Stress: contains information of 809 faecal samples, including:
    – the location of the sample
    – the time of sampling
    – the DNA-matched collared deer
    – the time when the deer was at the location

  • Reproduction Success: observations of 16 collared deer on:
    – if they were pregnant in one year
    – if they were accompanied by a calf in one year

The Data-Generating Process

  • A deer roams freely in the forest. Its movement is tracked by GPS collar.
  • A hunting event happens.
  • After some time, the deer defecates. This is a defecation event.
  • After some time, researchers go to the defecation location and collect an FCM sample. After their analysis, we know
    • to which deer the FCM sample belongs,
    • when (and where) the defecation event happened, and
    • the time delay between defecation and sampling.

Note: The defecation location is not the deer’s location at the time of the stress event.

Distance Approximation via Interpolation

TBD: Illustration.

Relevant Hunting Events

We introduce 4 Parameters:

  • Gut Retention Time (GRT) low [hours],
  • Gut Retention Time (GRT) high [hours],
  • Distance Threshold [km],
  • Proximity Criterion (“last”, “nearest”, or “higest score”)

A hunting event is considered relevant to an FCM sample, if

  • the time difference (“TimeDiff”) between experiencing stress (hunting) and defecation is in [GRT low, GRT high], and
  • the distance between the deer and the hunting event is \(\leq\) distance threshold at the time of hunting.

The Most Relevant Hunting Event

Among the relevant hunting events, the most relevant one is defined by the proximity criterion:

  • the closest in time (“last”),
  • the closest in space (“nearest”), or
  • the one with the “highest score.”

The scoring function is defined as TBD.

Illustration

TimeDiff Distance 14 hours 20 km 50 hours Number of otherrelevant huntingevent = 3 FCMSample Hunting events Nearest Highest score Last

  • GRT lower = 14 hours
  • GRT upper = 50 hours
  • Distance threshold = 20 km

The Fused Data

Extract Temporal Features

The Fused Data

Interpolate Movements

The Fused Data

Compute Spacial Distances

The Fused Data

Add Pregnancy Data

The Fused Data

Identify Events

The Fused Data

Finish Datasets

We suggest eight different Datasets for Modelling

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

The Models

For Modelling, we consider the following covariates:

  • Time difference [hours]
  • Distance [km]
  • Sample delay [hours]
  • Pregnant
  • Defecation day (between 1 and 366)
  • Number of other hunting events

The Models

Model Type Non-Parametric Effects Linear Effects Random Intercept Distribution Assumption
A GAM Time Difference, Distance, Sample Delay, Day of Year Pregnant, Number Other Hunts None Gaussian
B GAM Time Difference, Distance, Sample Delay, Day of Year Pregnant, Number Other Hunts None Gamma
C GAMM Time Difference, Distance, Sample Delay, Day of Year Pregnant, Number Other Hunts Deer Gaussian
D GAMM Time Difference, Distance, Sample Delay, Day of Year Pregnant, Number Other Hunts Deer Gamma

A Generalized Additive Model

  • \(FCM_i \sim \mathcal{N}(\mu_i, \sigma^2)\)

  • Identity Link: \(E(FCM_i) = \mu_i = \eta_i\)

  • Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_i = \beta_0 + \beta_1\,Pregnant_i +\\ \beta_2\,Number\,Other\,Hunts_i + f_1(Time\,Diff_i) + \\ f_2(Distance_i) + f_3(Sample\,Delay_i) + f_4(Day\,of\,Year_i) \end{gathered} \end{equation} \]

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

  • \(FCM_i \sim \mathcal{Ga}(\nu, \frac{\nu}{\mu_i})\)

  • For better Interpretability we use the Log-Link: \(E(FCM_i) = \mu_i = exp(\eta_i)\)

  • Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_i = \beta_0 + \beta_1\,Pregnant_i +\\ \beta_2\,Number\,Other\,Hunts_i + f_1(Time\,Diff_i) + \\ f_2(Distance_i) + f_3(Sample\,Delay_i) + f_4(Day\,of\,Year_i) \end{gathered} \end{equation} \]

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

B Generalized Additive Mixed Model

  • \(FCM_{i\,j} \sim \mathcal{N}(\mu_{i\,j}, \sigma^2)\)

  • Identity Link: \(E(FCM_{i\,j}) = \mu_{i\,j} = \eta_{i\,j}\)

  • Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_{i\,j} = \beta_0 + \beta_1\,Pregnant_{i\,j} +\\ \beta_2\,Number\,Other\,Hunts_{i\,j} + f_1(Time\,Diff_{i\,j}) + \\ f_2(Distance_{i\,j}) + f_3(Sample\,Delay_{i\,j}) + f_4(Day\,of\,Year_{i\,j}) \end{gathered} \end{equation} \] with: \(\gamma_j \overset{\mathrm{iid}}{\sim} \mathcal{N}(0, \sigma_{\gamma}^2)\)

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

i) Model specification

  • Gamma as distributional assumption, as FCM level is positive.

  • Log link for interpretability.

  • Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of FCM measurements for each deer.

    \[ \begin{eqnarray} \textup{FCM}_{ij} &\sim& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}) \\ \eta_{ij} &=& \beta_0 + \beta_1 \textup{Pregnant}_{ij} + \beta_2 \textup{NumberOtherHunts}_{ij} + \\ && f_1(\textup{TimeDiff}_{ij}) + f_2(\textup{Distance}_{ij}) + f_3(\textup{SampleDelay}_{ij}) + f_4(\textup{DefecationDay}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\sim& \mathcal{N}(0, \sigma_\gamma^2). \end{eqnarray} \]

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

Conclusion

  • Not many observations after datafusion left for robust modelling

  • Trade-off between spatial and temporal distance

  • Sample Delay seems to be significant

  • Modelling Outcomes don’t show much difference

  • Trade-off between Complexity and Explainability

Discussion

  • How to minimize spatial and temporal distance at the same time?

  • How to use a bigger Part of the Data?

Effect of Hunting on Red Deer